Constraints of this form on parameter spaces show up across a broad range of applications. In Molecular Dynamics, these types of manifolds are used when simulating the movement of atoms in molecules; the bond structure of these molecules determines a class of constraints, of the form in (REF ), on the distance and angles between atoms [1]}. The Bingham-von Mises-Fisher distribution, which can be derived by restricting a Gaussian distribution to a lower-dimensional unit hypersphere, exists on these types of manifolds. This distribution is of particular importance for applications to animal orientation, where the movements of animals are studied according to their directions with respect to a reference location [2]}. In computer graphics, constraint functions analogous to the function \(g\)  show up in texture synthesis, where the structure imposed by a sample image is learned and replicated [3]}. [4]} study the location of a human's joints to determine a 3D pose from 2D data; the location of these joints are constrained to exist at a certain distance determined by the known length of the person's limbs. In geology, these sorts of constraints are found in the study of crystallographic preferred orientations, where crystal samples are reoriented according to a known standard [5]}. Beyond these specific applications, classical statistical problems can also be re-expressed in terms of a constrained parameter space of the form in (REF ). A handful of these are used as proof-of-concept examples in this paper.
